1096
Ind. Eng.
Chem. Process Des. Dev. 1985, 2 4 , 1096-1104
Kullerud, G., personal communication, 1981. Lebedev, V. V. Khlmaya Tverdoga Topleva 1977, 1 1 (8), 11. Lee, J. M.; Tarrer, A. R.; Guln. J. A.; Rather, J. W. Prepr. Pap-Am. Chem. Soc., Div. Fuel Chem. 1977, 22 (6), 120. Mukherjee, D.K.; Chowdhury, P. B. Fuel 1965, 55, 4. Neavel, R. C. Fuel 1978, 55, 237. Semones. G. B. M. S.Thesis, Purdue University, West Lafayette, IN, 1982. Shalabi, M. A.; Baldwin, R. M.; Bain, R. L.; Gray, J. H.; Golden, J. 0. Ind. Eng. Chem. Process Des. Dev. 1979, 18(3), 474. Stach, E.; Mackowsky, M.-TH.; Teichmuller, M.; Taylor, G. H.; Chandra, D.; Teichmuller, R. “Stach’s Textbook of Coal Petrology”, 2nd ed.; Gerbruder Borntraeger: Berlin, 1975. Stephens, J. F. Fuel 1979, 5 8 , 489.
Tarrer, A. R.; Guln, J. A.; PMs, W. S.;Henley, J. P.; Prather, J. W.; Styles, G. A. I n “Liquid Fuels from Coal”; Ellington, R. T., Ed.; Academic Press: New York. 1977: D 45. Whitehwst, D. D.;MkC’hei,T. 0.; Farcasiu, M. “Coal Liquefaction”; Academic Press: New York. 1980:. Chaoters 1 -and 2. ~ r Wood, K. V.; Albrlght, L. F.;Brodbelt, J. S.;Cooks, R. G. Anal. Chim. Acta, in press. Yarzab, R . F.; Given, P. H.; Spackman, W.; Davis, A. fuel 1980, 5 9 , 81. ~
~~
Received for review January 4, 1984 Revised manuscript received January 14, 1985 Accepted February 13, 1985
Two-Bubble Class Model for Churn-Turbulent Bubble-Column Reactor Yatlsh T. Shah and Sebastlan Joseph University of Plttsburgh, Department of Chemical & Petroleum Engineering, Pittsburgh, Pennsylvania 1526 1
Dennls N. Smith’ and John A. Ruether Plffsburgh Energy Technology Center, U.S. Department of Energy, Pittsburgh, Pennsylvania 15236
A model for the bubble-column reactors operating in the churn-turbulent flow regime is developed. The overall conversion in the bubble-column reactor is related to the fractional throughput of large bubbles and small bubbles. The model parameters are either estimated from literature or measured. The fractional gas holdups and bubble rise velocities are measured by using a dynamic gas disengagement method. The volumetric mass-transfer coefficients for large and small bubbles for a variety of systems are either measured or estimated. Predictions of the two-bubble class model are compared with those of the conventional single-bubble class model. The results show that under a variety of conditions, the two-bubble class model gives substantially different results from those for a single-bubble class model. Further development of the two-bubble class model for use in design of bubble-column reactors is indicated.
In bubble-column reactors, the hydrodynamics, transport, and mixing properties, such as phase holdups, fluid-fluid interfacial areas, and interphase-transfer coefficients, depend strongly on the prevailing flow regime. Based on their studies, Schumpe et al. (1979) recommended the selection of a reactor geometry such that the reactor could be operated in the homogeneous flow regime. However, in large-diameter industrial bubble-column reactors, especially in cases where comparatively high superficial gas velocities (>0.1 m/s) are used to keep the catalyst uniformly suspended, the churn-turbulent regime might not be avoided. In fact, Bach and Pilhofer (1978) suggested that the churn-turbulent regime is the most commonly encountered flow regime in industrial bubble columns. Many investigators (Vermeer and Krishna, 1981; Schumpe, 1981; Godbole et al., 1982; Kelkar et al., 1983) have concluded that the bubble size distribution in the churn-turbulent regime can be approximated by a bimodal distribution, i.e., large bubbles and small bubbles. Thus, the gas holdup structure in the churn-turbulent flow regime consists of fast-rising large bubbles through a dispersion of uniformly sized small bubbles. Vermeer and Krishna (1981) have suggested that gas transport in the churn-turbulent regime occurs exclusively by large coalesced bubbles and neglected the role of mass transfer from 0196-4305/85/1124-1096$01.50/0
small bubbles. The design and scale up of bubble-column reactors operating in the churn-turbulent flow regime should be based on a model that considers the contribution of each bubble class. Also, the estimation of nonadjustable parameters, such as the liquid-side mass-transfer coefficient and the interfacial area using two-bubble classes, should be more realistic than the ones calculated from the average bubble size data. In the two-bubble class model, it can be assumed that the fast-rising large bubbles rise in plug flow. On the other hand, small bubble backmixing is more difficult to quantify, and a realistic model should take into account axial dispersion of the small bubbles. Plug flow for small bubbles is only realizable at very low gas velocities when the bubbly flow regime is present. As a limiting case in large diameter columns, the small bubbles are completely mixed. If sufficient data for the axial variations of the reactant in the small bubbles and the liquid phases are known, one can establish a relationship between the backmixing in the liquid phase and small bubbles. Vermeer and Krishna’s (1981) postulation about the negligible contribution of small bubbles to the gas-phase transport might be considered as an extreme case. The relative contribution of small and large bubbles to the gas transport and mass transfer can have a significant effect on the overall conversion in a bubble-column reactor operating 0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
-,I
material balance for the reactant A in the liquid phase can be expressed as
.,
/
1097
L LARGE BUBBLES
SMALL BUBBLES
LIQUID
tL_. t
c
It +
B U B B L E CLUSTER FORMATION
If the reaction is catalytic, the liquid-olid mass-transfer resistance is assumed to be negligible, and the kinetic constant K A includes the catalyst concentration in the slurry phase. The equilibrium liquid-phase concentration of component A for small and large bubbles will be assumed to follow Henry's law. Thus,
The gas velocity of the reactant is reduced due to chemical reaction. This variation in gas velocity can be related to the conversion by the expression
GAS
Figure 1. Schematic representation of a bubble column in the churn-turbulent flow regime (Vermeer and Krishna, 1981).
in the churn-turbulene flow regime. The purpose of the present paper is to develop a two-bubble class model for a bubble-column reactor operating in the churn-turbulent regime. The predictions of the two-bubble class model are compared with those of the conventional single-bubble class model. The analysis is carried out for a simple first-order slow reaction. The contribution of small bubbles to the overall mass transfer is evaluated based on the measurement of Sauter mean bubble size by the conductivity method. Development of the Model Consider a bubble-column reactor operating in the churn-turbulent flow regime (see Figure 1). As discussed above, in the churn-turbulent flow regime, the gas flow can be broken down approximately in terms of transport flow (i.e., flow by large bubbles) and entrained flow (i.e., flow by small bubbles) (Vermeer and Krishna, 1981). The large bubbles rise much faster through a swarm of small bubbles and can be assumed to rise in a plug flow manner (Vermeer and Krishna, 1981). The small bubbles are assumed to be completely backmixed. No interaction between the twobubble classes is considered. Assume a gas-liquid reaction (1) A(gas) V B ( , ~ ~products )
+
UG =
uG,O(l
+ aXA)
(6)
where UG(XA = 1) a=
XA =
UG(XA =
0)
UGWA= 0) ~ G , O Y A-GUGYAG ,O ~G,OYAG,O
(7) (8)
From eq 7 and 8,
UG =
U G , O ( ~ A+G~, OY A G , o ) (YAG,O +~YAG)
(9)
The flux of A through large bubbles and small bubbles is given by uGL
=
Ub>tGL
uGs
=
Ub:cGS
(10)
Now eq 2, 3, and 4 simplify as
-
If the reaction is noncatalytic, it is assumed to be slow so that it occurs in the bulk liquid. If the reaction is catalytic, the reaction occurs at the catalyst surface in the liquid phase, and the catalyst is assumed to be uniformly distributed in the liquid phase. The rate of the reaction is only dependent on the liquid-phase concentration of component A (i.e., the concentration of species B is in excess), and it is assumed to be first order. The pressure in the reactor is assumed to be constant. With these assumptions, a mass balance for the reactant A in large bubbles yields
The mass balance for the reactant A in the small bubbles can be expressed as RT U G S ( Y A G , O S - y A G , l s ) = L ( k L d S p [ ( C ~ * ) S- C A L I (3) The liquid phase is assumed to be completely backmixed and to be operated in a batch (i.e., no flow). Thus, the
P
E
Ubr,OScG,OSYAG,OS
-
P
ubr,lStG,lSYAG,lS
(I2)
Since the concentration of A in the large bubbles varies along the length of the column, an average value of the driving force is used in the well-mixed liquid-phase balance. Using the variation of gas velocity along the column, given by eq 9, a mass balance for reactant A in large bubbles is obtained:
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
1098
The mass balance for reactant A in small bubbles can be related to the liquid-phase concentration as
CAL = -
~~br,Os~G,OsyAG,Os(
-
yAG,Os
‘AG>:
+-P Y A G s
+~YAG
LRT(kLa)’
H
Thus, eq 14 can be rewritten as
where
The boundary conditions for eq 16 are YAGL
=
YAG,oL
at z = O
(17)
The liquid-phase concentration of reactant A is constant throughout the reactor, and hence eq 16 can be integrated analytically as z=
A’(ymL- YAG,o~) (1 +
a)YAG,OL(YAG,OL
B’
- In
+
+ (.YAGL)
Y A G , O+~~ Y A G cr ~
a
YAG,oL(1
In + a) + b
a -~YAG,o~ (18) a - bYAGL
where
bB C’= a
CALL(kLa)LR7‘
a =
ubr,0LtG,0L(YAG,0L)2(1
b=
+ a)p
L(kLa)LRT U b r , ~ L c ~Y, A ~L G(, O ~ )+~ (a)H ~
From the liquid-phase balance, (kLa)SPYAGS H
CAL=
+
(kLa)zyAGL dz
+
( k ~ a ) (kLa)L ~ + kAeL
(19)
Equations 15, 18, and 19 have to be solved simultaneously to obtain the values for YAGs and the profile for YAGL. In order to calculate the gas-phase conversions for component A, the mass balance for the gas phase at the outlet of the reactor can be written as Ub?EGSYAG,lS + UbrLtGLYAG,lL= UGyA,l
(20)
or Ubr,OS%,OSYAG,OS yAG,Is
Ubr,OLeG,OLYAG,OL YAG,lL (YAW’ + ~ Y A G , ~ ’ ) (YAG,oL + a Y A G , i L ) UG,OYAG,OYAGJ (21) (YAG,O +~YAG,~)
and
+
Thus, the overall conversion for the first-order reaction in the churn-turbulent regime can be calculated if the fractional gas holdups, the rise velocities, and the volumetric mass-transfer coefficients for two-bubble classes and the kinetic data are known. For uniform-sized bubbles, with the assumption of plug flow in the gas phase and of a well-mixed liquid phase, the conversion of reactant A can be calculated from
The conversion, X A ,as a function of kLa, kA, and eG (Le., 1 - eL), can be obtained by integrating eq 24 with the boundary condition YAG = YAG,O at z = 0. The integration procedure is the same as before and uses eq 6,22, and 24. The final calculation requires a trial-and-error procedure for CALso that both eq 23 and 24 are simultaneously satisfied.
Model Parameter Estimations The two-bubble class and single-bubble class models described above contain a number of independent hydrodynamic and transport parameters, such as eGL, eGS, tG, (kLa)L, and kLa, that will determine the performance of the reactor. The total gas holdup tG depends on the gas velocity, liquid properties, solid particle size, concentration, density, etc. The best available literature correlations for tG for various systems are described in Table I. Recently, numerous studies have been reported for the measurements of eGL and eGS for a variety of systems using the dynamic disengagement technique. The results of these studies are summarized in Table 11. The reported correlations for kLa are summarized by Shah et al. (1982). The parameter can be divided into two parts, and ecL,where (a’)L= u ~ / ~ The G ~ reported . studies indicate that in churn-turbulent flow, (kLa’)Lis independent of gas velocity and depends largely upon the fluid properties. Several published data for (kLur)Lare summarized in Table 111. The parameter (kLa)Scan also be = divided into two parts, (kLa’ISand tGS,where aS/tGs. Both kLS and (a’)s depend upon fluid properties and gas velocity. The effects of fluid properties, gas velocity, and bubble diameter on kLScan be estimated from the study of Calderbank and Moo Young (1961). Some other reported studies are also summarized by Shah et al. (1982). The parameter is related to the Sauter mean bubble diameter dv: as follows: = 6/dV:. This in turn depends upon the gas velocity and fluid properties. In the present study, both (d.JL and (dJS were measured by using a conductivity probe. This experimental study is briefly described below. Measurement and Interpretation of Bubble Size An electrical conductivity probe device was developed to measure local bubble size and velocity of the gas phase in a three-phase slurry bubble column. The slurry bubble-column apparatus is shown in Figure 2. The bubble column is a transparent plastic cylinder having an inside diameter of 10.8 cm and a length of 194 cm. A multipleorifice plate (76 X 0.1 cm triangular pitch) located at the
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
Table I. Reported Gas Holdup Correlations for Various Systems system range of params 1
ref
‘G/(I - CGl4 = C(~D:PL/U)’/~ (gD,3)’i’2
UG, m/s: 0.003-0.4
air-water
correlat proposed
1099
Akita and Yoshida, 1973
VL2
air-glycol aq soln air-methanol He-water Cop-water
UL, m/s: 0.0-0.044 D,, m: 0.152-0.6
uG/ [
p L , kg/m3: 800-1600 p L , P a s: 0.000 58-0.021 u, N/m: 0.022-0.074
C = 0.2 for pure liq and nonelectrolytes C = 0.25 for electrolytes
air-ethanol-solids air-aq glycerol-solids air-methanol-solids air-water-solids
U G , m/s: 0.05-4.0 D,, m: 0.05 and 0.1 d,, m: 0.011-0.0287 Him m: 0.05-0.2
~DC)”~]
2 Kit0 et al., 1976
pL, kg/m3: 790-1210 p L , pa s: 0.001-0.062
u, N/m: 0.0233/0.0728
3 air-alcohols air-halogenated hydrocarbons
UG,m/s: 0-0.1 D,, m: 70.1 Hz, m: 7.12
Bach and Philofer, 1978
air, HP, Cop, CHI C3H8,Nz-water air-org liq air-electrolyte soln.
UG, m/s:
Hikita et al., 1980
D,, m: 0.1 H2 m: 0.65
4 0.042-0.38
p L , kg/m3: 790-1170 p L , P a s: 0.009-0.0178 u, N/m: 0.0229-0.0796 pG,
Z = ionic strength of the soln
kg/m3: 0.84-1.84 Gas vent
G-Ll
Sample
3 m m stainless steel tubing and compression fitting with Teflon seal D.C.+
Sample Temaercture indicator
recirculation
Thermocouple
Sample
T e m p era! ure controiler
0
Bubble c o l u m r
Heater
Sample port
D.C.+ 0.076mm-diometer chrome1 probes
Ceramic support
P t tT t t T t
Slurry/gos mixture flowing cocurrently upwards
1
P
l I u
I
1
Figure 2. Schematic diagram of 10.8-cm4.d. slurry bubble-column apparatus.
bottom of the column is used to introduce the slurry and gas phases. Several ports are located along the axis of the column to allow the insertion of the probe. A depiction of a twin-electrode conductivity probe inserted into the slurry bubble column is given in Figure 3. The probe consists of two Teflon-coated wires with a diameter of 0.076 mm. The Teflon coating serves as a hydrophobic surface for rapid shedding of the liquid by the bubbles and as an electrical insulator. Chrome1 wire has proven to be a good compromise between strength and electrical conductivity in a slurry environment. A pair of wires is threaded through ‘Is-in. tubing and a two-hole ceramic cylinder having an outside diameter of 1.6 mm. The ceramic cylinder is bent perpendicular to the tubing and sealed to the tubing with epoxy. The Teflon wire is cut with a scalpel approximately 1 mm below the ceramic cylinder to ensure a minimal exposed area of wire. A
Figure 3. Probe configuration for conductivity measurements in 10.8-cm-Ld. slurry bubble-column apparatus.
Conax fitting with Teflon seals is fitted over the tubing through a port in the column wall and facilitates the radial placement of the probe. The radial positioning of the probe is accomplished with a calibrated ruler and a reference point located outside the column. The probe circuit consists of a variable dc power supply, probe, wave-form recorder, junction box, and doublepole-double-throw switch. The junction box is used to select any one of three probes installed in the bubble column. A double-pole-double-throw switch is used to reverse current occasionally to dispose of any charge that may be acquired on the probe tip. The wave-form recorder is an analog-to-digitalconverter with a solid-state memory that stores the digital equivalent of an analog electric signal. The model that was used in our test facilities, a BIOMATION model 2805 M, provides for the simultaneous recording of two channels with sampling frequencies up to 5 MHz. The solid-state memory provides space for 2048 numbers for each channel with a resolution of 1part in 256. The contents of the memory, representing the recorded traces, are displayed on a CRT
1100
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
Table 11. Large and Small Bubble Gas Holdups as a Function of Gas Velocity for Various Systems u,, m/s cg cgS cgL system air-water 0.0787 0.1534 0.1135 0.0399 0.118 0.1881 0.1333 0.0548 Godbole et al., 1984 0.2241 0.158 0.0661 0.154 0.193 0.2474 0.170 0.074 air-soltrol-130 0.111 0.280 0.234 0.046 Godbole et al., 1984 0.137 0.298 0.251 0.047 0.164 0.318 0.262 0.056 0.201 0.338 0.274 0.064 0.235 0.354 0.280 0.074 0.110 N2-turpentine-5 0.277 0.197 0.08 0.165 0.336 0.228 0.108 Vermeer and Krishna, 1981 0.218 0.382 0.248 0.134 0.272 0.418 0.260 0.158 0.324 0.45 0.271 0.179 air-0.5 wt % CMC s o h 0.0484 0.0658 0.0398 0.026 0.0933 0.0976 0.0516 0.046 0.1345 0.0625 0.072 0.140 Godbole et al., 1983 0.186 0.1692 0.0802 0.089 0.237 0.1798 0.0797 0.1001 0.280 0.1979 0.0819 0.116 air-water + 10 wt % 0.0478 0.1172 0.0882 0.029 polystyrene 0.0918 0.1614 0.1164 0.045 Godbole et al., 1983 0.1912 0.1192 0.072 0.14 0.187 0.2172 0.1422 0.075 0.2372 0.1532 0.084 0.230 0.2533 0.1603 0.093 0.246 0.0499 0.1131 0.0761 0.037 air-water + 20 wt % polystyrene 0.1844 0.1204 0.064 0.140 0.1951 0.1141 0.081 0.186 Godbole, 1983 0.2368 0.1388 0.098 0.233 0.258 0.2570 0.133 0.124 0.0483 0.1070 0.066 0.034 air-water + 30 wt % polystyrene 0.0943 0.1364 0.082 0.054 0.2074 0.1084 0.099 0.186 Godbole, 1983 0.233 0.2131 0.106 0.107 0.2409 0.111 0.130 0.258
r----: cgL/cg
0.2601 0.2913 0.295 0.3128 0.1643 0.1577 0.1761 0.1893 0.2090 0.288 0.321 0.351 0.378 0.398 0.395 0.471 0.535 0.526 0.556 0.586 0.247
air-water Godbole et al., 1984
air-soltrol- 130 Godbole et al., 1984
Nz-turpentine-5 Vermeer and Krishna, 1981
air-0.5 wt % CMC soln Godbole et al., 1983
air-water
+ 10 wt %" polystyrene
Godbole et al., 1983 (I
U,, m/s kLa, s-I 0.0787 0.118 0.154 0.193 0.229 0.106 1.30 0.156 0.191 0.223 0.110 0.165 0.218 0.272 0.324 0.0484 0.0933 0.14 0.186 0.237 0.280 0.0478 0.0918 0.14 0.187
0.0578 0.0774 0.951 0.1103 0.1176 0.08 0.0906 0.1011 0.1142 0.1254 0.0434 0.0594 0.0737 0.0866 0.10 0.00289 0.00531 0.01 0.01 0.0126 0.0147 0.029 0.050 0.069 0.074
Bubble Column
Display Power Computer
Figure 4. Two-point electrical conductivity probe circuit and data acquisition system.
5 0 0
5
loo[ 50
02
0.396 0.477 0.502 0.539
Table 111. Comparison of the Volumetric Mass-Transfer Coefficients for Large Bubbles with the Assumption of Gas Transport Occurring through Large Bubbles Only system
L----J
0.279 0.376 0.345 0.354 0.367 0.333 0.347 0.415 0.414 0.482 0.317
( k L a ) /egL s-l
1.45 1.41 1.44 1.43 1.22 1.74 1.93 1.80 1.78 1.69 0.53 0.55 0.55 0.55 0.56 0.11 0.115 0.112 0.112 0.126 0.126 1.0 1.11 0.96 1.0
kLa values estimated.
monitor. In addition, the collected information is transmitted to a computer for analysis via a digital output
I
I
04
06
08
I O
12
TIME, seconds
Figure 5. Typical trace of conductivity probe response with time.
connection. Figure 4 depicts the probe circuit and data acquisition system. For a typical run, a series of scans under identical conditions is taken to obtain a reasonably large statistical sample. The sampling interval for most of the test is 0.5 ms per point, and the total sample time is nominally 60 s for each channel. Figure 5 shows a plot of the data generated with one scan. The signal intensity is obtained as a function of time and can be related to bubble characteristics, such as bubble velocity, bubble size, and gas void fraction. In order to interpret the conductivity probe signals, the following computations are made. The slope of the signal intensity is computed continuously, and when the slope exceeds a threshold positive value, the start of a bubble signal is indicated and stored. The end of a bubble signal, corresponding to the bubble leaving the probe, is indicated and stored when the slope for two consecutive points changes from less than a threshold negative value to greater than a threshold negative value. The lag time, At, between a pair of signals, corresponding to the passing of a bubble, is computed from the time interval between the centroid of the two bubble responses. The centroid of a bubble response is obtained from the dwell time, 7,of the bubble on a probe and the initial time, t , that the bubble strikes the probe. Thus, At = (tz - ti) =
- 71)
1/2(72
(25)
The effect of taking the centroid of the pierced bubble as the reference frame for the computation of lag time is to average small deviations in dwell time associated with the passage of the bubble through the probe tips.
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 1101
The matching process of a signal pair representing a bubble passing through both probes requires several conditions to be satisfied. A discussion of the scenarios for improper matching of a pair of bubble responses has been given by Serizawa et al. (1975). In order to avoid improper matching of a signal pair, three constraints are placed on the matching process. The dwell time of a pair of signal responses must be similar (less than 30% relative deviation). The velocity of all bubbles must be greater than the velocity of the smallest measured bubble length as calculated from Stokes law and less than the velocity of a bubble slug bridging the column diameter calculated from the relationship given by Nicklin et al. (1962). The ascendingvelocity, ub, and length, A, of each bubble passing vertically upward through the probes can be calculated from the lag time, dwell time, and vertical gap between the twin probes. Thus, u b = h/At (26) X = hr/At
(27)
A comparison of the measured and calculated bubble length distributions indicates very good agreement. The cumulative bubble size distribution is derived from integration of eq 29 as CX(db) = 0.5 + 0.5 erf [In (db/dbg)/d?1/2] (35) The cumulative volume distribution is derived from the third moment of the log-normal distribution CV(db) = 0.5 + 0.5 erf [(In (db/dbg) - 3 ~ ~ ) / ~ 2(36) ~’~] The Sauter mean bubble diameter is the ratio of the third to second moment of the log-normal distribution function d,, = dbg e x p ( 2 . 5 ~ ~ )
The Sauter mean bubble diameter may also be expressed in terms of the mean bubble length d,, = 1.5X
= t,/t,
(28)
Interpretation of bubble length and velocity distributions was performed with the probability model developed by Tsutsui and Miyauchi (1980) and applied to conductivity probe measurements by Hess (1983). Two distribution functions, log-normal and y, have been considered to describe the bubble size distributions measured with the conductivity probe; in all cases, the log-normal distribution gives a better match to the measured distributions. The log-normal distribution of bubble sizes may be expressed as X(db) = exp(-([ln (db/dbg)12/2u2)~/(2.)1’2udb
(29)
In general, the measured bubble length is not equal to the bubble diameter (also called bubble size) but rather is associated with the probability of a single bubble striking the probe over the projected area of the bubble. The necessary correlations between bubble length and bubble size distributions have been derived by Hess (1983). These can be expressed as
[8/S2(X)/X+ 11
u2 = In
(30)
(38)
The mean bubble diameter is obtained from the first moment of the log-normal distribution
The gas holdup is calculated from the ratio of the total time of bubble responses to the total sample time. eg
(37)
db
= dbg exp(u2/ 2)
(39)
For the two-bubble class model, a Sauter mean bubble diameter must be calculated for the small and large bubbles. The Sauter mean bubble size up to a bubble size db, (db’), is derived from the third and second moments. dv,(dbs) =
0.5 + 0.5 erf ([In (dbS/dbg) - 3 ~ r ~ ] / a 2 ~ / ~ ) dbg e x p ( 2 . 5 ~ ~ ) 0.5 + 0.5 erf ([ln (dbS/dbg) - 2u2]/u21/2) (40) The Sauter mean bubble size for bubbles larger than db, (dbL),is a function of the cumulative bubble volume, Sauter mean bubble size for all the bubbles, and the Sauter mean bubble size for bubbles less than dbs:
The gas holdup fraction related to the small and large bubble may be expressed in terms of the total gas holdup fraction and cumulative bubble volume: (42)
and dbg = 1.5X exp(-2h2)
(31)
where n
X
Xi/N
= i=l
n
S2(X) =
r=l
(Xi - X)2/(N- 1)
(32)
From the probability model, the cumulative bubble length distribution can be expressed in terms of the two log-normal distribution parameters, u2 and dbg, as CZ(X) = 0.5(X/dbg)2exp(-2a2) erfc [In (X/dbg)/~T2~/~] + 0.5 {I+ erf [(In (X/dbg) - 2 ~ ~ ) / & / ~(33) ])
A measured cumulative bubble length distribution is obtained by sorting the bubble lengths in ascending order. The measured cumulative bubble length distribution may then be expressed simply CZ(XJ = i / N (34)
€gL = €g - €gs
(43)
The above equations will give all the required quantities to interpret the bubble dynamics for a two-bubble class model. Results for Bubble Size Distribution The experimental results of the bubble size distribution as a function of cumulative gas fraction and gas velocity for N2-water and N2-Ater-10 wt ‘70glass beads systems are described in Figures 6 and 7, respectively. The results show several interesting but expected effects. In the bubble flow regime (og5 0.09 m/s), an increase in gas velocity somewhat reduces the Sauter mean bubble size. In the highly churn-turbulent flow regime CogI 0.15 m/s), on the other hand, the Sauter mean bubble size remains essentially constant. The maximum changes occur between the gas velocities of 0.09 and 0.15 m/s. This change may be caused by the change in the flow regime. This sudden increase in Sauter mean bubble size with increasing gas velocity is illustrated in Figure 8. Consistency in mass balance requires that this increase in Sauter mean bubble
1102
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
o o
I O
09
0 016
z
2
F
08
0
07
E w
06
W
2
w
9 w A
m m
7
i
05
g 0010
04
9z
m 5 E
w
03
3
a
Lo
0 2
0
,
01
0
I ::::I 0.000
5'
0006
a
A
23
0012
001
0 03 005 BUBBLE SIZE, m
007
0
Figure 6. Cumulative bubble volume fraction as a function of bubble size for N2/H20 system.
,
I
003 0.09 0 15 SUPERFICIAL GAS VELOCITY, m/s
0 21
0
Figure 8. Sauter mean bubble size and average bubble dwell time system). as function of superficial gas velocity (N2/H20 0.01 2 1
09
II'
1
I
I
I
0.0lOt
I
I
1
0
e
v)
0
1
e
e
i
4 0
3
a
03
Symbol
0.002
0
System
Nitrogen/water Nitrogen/water/lO wt ?4 solids
0.03 0.06
0.09
0.12
0.15
0.180.21
S U P E R F I C I A L GAS V E L O C I T Y , m/s
Figure 9. Sauter mean bubble size for large bubbles as a function of superficial gas velocity.
BUBBLE SIZE, m
Figure 7. Cumulative bubble volume fraction as a function of w t % glass beads (mean particle diameter bubble size for N2/H20/10 = 48.5 km) system.
size should be accomplished by a drastic reduction in the number of bubbles. If this were the case, the dwell time (which+is inversely proportional to the frequency of the bubble passage across the probe) would also show a sudden increase at the flow transition. In the present study, for N,-water system, increasing the gas velocity from 9 to 15 cm/s, decreased the bubble frequency, f , from 20.5 to 13.1 s-l. Figure 8 also illustrates the average dwell time of the bubbles on the probe as a function of superficial gas velocity for the N2-water system. The average dwell time, T , is defined as i== t g / f
(44)
Increasing gas velocity from 0.09 to 0.15 m/s nearly doubled the average dwell time. This indicates that the bubble coalescence rate increased sharply with increasing gas velocity during the flow transition. Visual observations also indicated this to be the case. The presence of solids only mildly affects the bubble size. The fluid properties (such as surface tension and viscosity) should, however, have a pronounced effect on the bubble size. The bubble
size distribution was measured at a number of different radial positions. The radial volume balance on the gas phase based on the bubble size distribution gave us a closure and good agreement with independently measured %
The dynamic disengagement technique described earlier was used to define the breakpoint diameter, dbS,between large and small bubbles (Godbole et al., 1982; Godbole et al., 1984). The values for eGL and eGS for air-water and air-water-10 wt % glass bead systems that are shown in Table I1 were used in interpreting the bubble size distribution data. From the results shown in Figures 6 and 7 using the above described equations, one can obtain a Sauter mean bubble diameter for large (dVsL) and small (d,,:) bubbles. The calculated dv: and dv: as functions of gas velocity for Nz-water and Nz-water-10 wt % glass bead systems are shown in Figures 8 and 9. The results are interesting. In the homogeneous flow regime, the results are in agreement with those reported in the literature (Akita and Yoshida, 1973). In the churn-turbulent flow regime, both dWSand dWLincrease with gas velocity largely due to increased coalescence. Both dv: and dvsLshould also depend on the fluid properties. More data are needed with a variety of fluids.
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 ,081
Table IV. Conversions Predicted by Single- and Two-Bubble Class Model for Air-Water-Type System (a = 0.0)
UG,m/s 0.09 0.12 0.15 0.20 0.25 0.30
I
I
I
I
1
1103
I
I
XSBC XTBC XSBC XTBC XSBC XTBC 0.409 0.369 0.346 0.288 0.243 0.209
0.478 0.411 0.378 0.309 0.258 0.220
0.481 0.457 0.450 0.389 0.339 0.298
0.585 0.522 0.509 0.434 0.374 0.326
0.551 0.547 0.570 0.522 0.475 0.430
0.694 0.659 0.650 0.592 0.535 0.484
Table V. Conversions Predicted by Single- and Two-Bubble Class Model for Air-Water-Type System (CY= -0.5)
c"
N i t r o g e n /water N i t r o g e n / w a t e r / l O w t % solids
.01
a v)
0
0.09
0.03
UG,m / s 0.09 0.12 0.15 0.20 0.25 0.30
XSBC XTBC XSBC XTBC XSBC XTBC 0.452 0.406 0.377 0.309 0.260 0.220
0.547 0.470 0.426 0.342 0.281 0.237
0.541 0.511 0.502 0.431 0.371 0.322
0.671 0.607 0.580 0.490 0.417 0.360
0.625 0.620 0.646 0.591 0.533 0.478
0.786 0.749 0.743 0.674 0.606 0.545
Model Evaluation In the present study, the predictions of the conversion of the gaseous reactant A by the two-bubble class model are compared with those of a single-bubble class model. For the sake of illustration, the hydrodynamic data for the N2-water type of system are used. The backmixing of both the small bubbles and the liquid phase is assumed to be complete. The conversion for a slow first-order gas-liquid reaction or a first-order gas-liquid-solid reaction would depend on the independent parameters e,, , :e e s, kA, (kLa)L,( ~ L U ) ~ , Ug, and H. The parameters kA and dpertain to the given reaction system. In the present analysis, kA values of 0.075, 0.15, and 0.5 s-l and H = 25.526 atm m3/(kg mol) were assumed (Danckwerts, 1970). This range of kA values covers the practical cases, such as absorption of COz in water and others outlined by Danckwerts (1970). The hydrodynamics and transport properties of the reaction system were assumed to be similar to those of air-water. For this system, the dependences of E e and :e on gas velocity are described in Tables I ancf'Il!, hnd the dependences of kLa and ( ~ L u ' )on~ gas velocity are described in Table 111. The value of (k# was estimated from Calderbank-Moo Young's correlation to be 0.04 cm/s independent of gas velocity. The value of a'as a function of gas velocity was estimated from the Sauter mean bubble size dv: data reported in Figures 8 and 9. For completely backmixed small bubbles, the predictions of X A as a function of U, for a few typical values of kA by both models are illustrated in Tables IV and V for CY = 0 and -0.5, respectively. Values of CY = 0 and -0.5 imply 0% and 50% gas-phase contraction, respectively. These results indicate substantial differences in the predictions of XA by single-bubble class and two-bubble class models particularly at low gas velocities (0.09 m/s < U, < 0.20 m/s). For both CY = 0 and -0.5, the conversions predicted by the two-bubble class are always higher than those predicted by the single-bubble class model. I t must be emphasized, however, that a 50% contraction in gas volume would result in an axial dependence of the gas holdup, volumetric mass-transfer coefficients, etc., and the equations developed for the single- and two-bubble classes should include the axial dependence of these parameters. Future study must consider this added complication. The magnitude
0.21
0.15
SUPERFICIAL GAS VELOCITY,
m/s
Figure 10. Sauter mean bubble size for small bubbles as a function of superficial gas velocity. I C
1
'
1
'
1
'
-Two bubble class model ---
0 8
S i n g l e b u b b l e class model
a
=X
0 6
0 v)
LL
w
>
5
04
0
02
I 005
oc
I 015
,
I
025
0
SUPERFICIAL GAS VELOCITY, U g (m/s)
Figure 11. Effect of k A = 0.15 s-', CY = 0.0.
on conversion (two-bubble class model),
of the difference in predicted conversion according to the two- and single-bubble class models for CY = -0.5 and 0 indicates that for reactions with appreciable change in gas flow, the use of the more complicated model is justified. When :e 0, the predictions of the two-bubble class and single-bubble class models coincide. Furthermore, when 0, predictions by the two models will be close to each other. The discrepancy in the predictions by the two models is, therefore, due to the mass-transfer contribution of small entrained bubbles. For a typical value of k A = 0.15 s-l, the effect of ( k ~ aon) the ~ prediction of X A by the two-bubble class model is described in Figure 11. The results clearly indicate that an increase in (kLa)S initially increases XA, but after a certain critical value, XA becomes independent of (kLa)S.If the backmixing of small bubbles was not complete, the predictions of XAby the two-bubble class model would be further increased. Predicted conversions with both the single- and twobubble class models depend strongly on the values of hydrodynamic and transport parameters. Our calculations indicate that the results for the air-water-10 wt % glass bead system would be similar to those described above for the air-water system. For the air-0.5 wt % CMC solution system, the two-bubble class and single-bubble class model predictions would be much closer, largely due to the
-
1104
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
Table VI. Conversions Predicted by Single- and Two-Bubble Class Model for Air-0.5 wt % CMC Solution-Type System (CY = 0.0, k A = 0.15 s-’) UG,m/s
XSBC
XTBC
0.09 0.14 0.19
0.099
0.114 0.110
0.24
0.28
0.101
0.093 0.081 0.078
0.099 0.086
smaller contribution by entrained bubbles to the overall mass transfer in this system. Some typical results for this system are shown in Table VI. The analysis presented here extends that proposed by Vermeer and Krishna (1981) to a reacting system. It is clear that the parameters involved in the two-bubble class model, namely (kLa)Sand (kh)L as well as :6 and tGL, are dependent upon the physicochemical properties (i.e., surface tension, viscosity, foaming capability, etc.) of the system in the same manner as both tG and kLa in the single-bubble class model. However, the present paper indicates a strong design basis for use of a two-bubble class model to describe bubble-column reactors with a wide bubble size distribution. It has been shown that realistic reaction conditions exist in which the new model and the conventional single-bubble size model predict substantially different conversions. Further development of the twobubble class model for use in reactor design is expected to yield an improved design method. More investigation of the two-bubble class model for a bubble column operating with a wide variety of physicochemical properties is needed.
Acknowledgment The help of Dr. S. P. Godbole with the mathematical modeling and the constructive discussions with Dr. N. L. Carr are gratefully acknowledged. The financial aid of Gulf Research and Development Co. for S. Joseph is also gratefully acknowledged. Reference in this report to any specific commercial product, process, or service is to facilitate understanding and does not necessarily imply its endorsement or favoring by the US Department of Energy.
Nomenclature gas-liquid interfacial area, m-l A = gas-phase reactant in the reaction given by eq 1 B = gas-phase reactant in the reaction given by eq 1 C = product formed in the reaction given by eq 1 CAL = liquid-phase concentration of component A, kg mol/m3 CAL* = equilibrium concentrationof component A in the liquid phase, kg mol/m3 CV(db) = cumulative bubble volume distribution CX(db) = cumulative bubble size distribution CZ( A) = cumulative bubble length distribution db = bubble size, m dbg= geometric mean bubble size, m d,, = Sauter mean bubble size, m f = bubble frequency, s-l g = acceleration due to gravity, ms-2 h = vertical distance between twin probes H = Henry’s constant, Pa m3/(kg mol) k A = kinetic constant for the first-order reaction, s-l a =
k ~ =a volumetric mass-transfer coefficient based on the dis-
persion volume, s-l
kLa’ = volumetric mass-transfer coefficient per clean liquid volume, s-l KA = overall absorption reaction coefficient, s-l L = length of the reactor, m n = number of bubble signals that are matched P = pressure, Pa R = gas constant, Pa m3/(kg mol K) S2(X) = variance of bubble length, m2 t = time at which probe pierces bubble T = temperature, K ribl = bubble rise velocity, m/s Ub = mean bubble velocity, m/s UG = superficial gas-phase velocity, m/s XA = gas-phase conversion of reactant A X(db) = frequency distribution of bubble size x = reactor distance, m YAG = gas-phase mole fraction of component A z = dimensionless reactor distance Greek Letters A t = log time, s X = bubble length, m T; = mean bubble length, m u2 = variance of bubble size t G = gas holdup tL = liquid holdup i= average bubble dwell time, s Subscripts 0 = inlet of reactor 1 = outlet of reactor Superscripts L = large bubble S = small bubbles
Literature Cited Aklta, K.; Yoshkia. F. Ind. Eng. Chem. Process Des. Dev. 1973, 12, 76. Bach, H. F.; Piihofer, T. Ger. Chem. Eng. 1978. 1 , 270. Calderbank, P. H.; Moo-Young. M. Chem. Eng. S d . lS81, 16, 39. Danckwerts, P. V. “Gas-Liquid Reactions”; McGraw-Hill: New York, 1970. Godbole, S. P. Ph.D. Thesis, University of Pittsburgh, Pittsburgh, PA, 1983. Godbole. S. P.; Honath, M. F.; Shah, Y. T. Chem. Eng. Commun. 1982, 16, 119. Godbole, S. P.; Joseph, S.: Shah, Y. T.; Carr. N. L. Can. J. Chem. Eng. 1984, 62, 440. Godbole, S. P.; Schumpe, A.; Shah, Y. T. Chem. Eng. Commun. 1983, 2 4 , 235. Hess, M. Final Report DE-4P22-83PC10614, July 1983. Hikta, M.; Asai, S.;Tanlgawa, K.; Segaur, K.; Kitao, M. Chem. Eng. J . 1980, 20, 59. Kelkar, B. G.; Godbole, S. P.; tionath, M. F.; Shah, Y. T.; Carr, N. L.; Deckwer, W. D. AICHE J. 1983, 29 (3), 361. Kito, M.; Shimida, M.; Sakai, T.; Sugiyama, S.; Wen, C. Y. Nuidization 1978, 411. Nicklin, D. J.; Wilkes, J. 0.; Davidson, J. F. Trans. I n s t . Chem. Eng. 1982, 40, 61. Schumpe, A. Doctoral Thesis, Universitat Hannover, Federal Republic of Germany, 1981. Schumpe, A,; Serpemen, Y.; Deckwer, W. D. Ger. Chem. Eng. 1979, 2 , 234. Serizawa, A.; Kataoka, I.: Michiyoshi, I . Int. J. Muniphase Flow, 1975, 2 , 221. Shah, Y. T.; Kelkar, 6 . G.; Godbole, S. P.; Deckwer, W. D. AIChEJ. 1982, 2 8 , 353. Tsutsui, T.; Miyauchi, T. I n t . Chem. Eng. 1980, 2 0 , 386. Vermeer, D. J.; Krishna, R. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 475.
Received for review January 18, 1984 Revised manuscript received December 26, 1984 Accepted February 25, 1985
